Based on the Collins integral formula, an analytical propagation equation of three-dimensional (3D) elegant Hermite-Gaussian (H-G) beams through a 4 × 4 paraxial optical system is derived, by means of which a detailed study of astigmatic H-G beams propagating through a three-cylindrical-lens mode converter is made. It is found that astigmatic elegant H-G beams can be converted to symmetric elegant H-G beams in the sense of second-order moments and the symmetric behavior remains upon propagation in free space, but this type of symmetrization is imperfect.